@article{c66232e7055f4ada9deff5337f776e6d,
title = "Absolute Galois Groups of p-Adically Maximal PpC Fields",
abstract = "We characterize the absolute Galois groups of the p-adically maximal PpC fields. As an application we solve affirmatively the (finite) inverse Galois problem over PpC fields with infinitely many (non-equivalent) p-adic valuations.",
author = "Ido Efrat",
note = "Funding Information: The following realization problem is typical in Galois theory: given a class of fields with specified arithmetical properties, what are the absolute Galois groups of its members? In this work we solve the realization problem for the class of the, so called, maximal pseudo /?-adically closed fields. These fields were recently investigated by Grob [G], mainly from the model-theoretic point of view. More specifically, we say that a field K is pseudo p-adically closed (abbreviation - PpC) if each non-empty absolutely irreducible affine variety defined over K has a /^-rational point, provided that it has a simple rational point in each /?-adic closure of K. A PpC field A^is called p-adically maximal if it has no proper algebraic extension L such that each /?-adic valuation on K extends to a ^-adic valuation on L. The dual group-theoretic notion (in the sense of the Galois correspondence) is that of a G(Q^-minimal group on a Boolean (= Hausdorff, compact and totally disconnected) space X. In the simplest case, in which X is the discrete space Xe of l < e < oo elements, this is defined {\"a}s The content of this paper corresponds to a part of the author's Ph. D. thesis carried out in Tel Aviv University under the supervision of Prof. Moshe Jarden. The work was partially supported by a grant from the G. I. F., the German-Israeli Foundation for Seientific Research and Development.",
year = "1991",
month = jan,
day = "1",
doi = "10.1515/form.1991.3.437",
language = "English",
volume = "3",
pages = "437--460",
journal = "Forum Mathematicum",
issn = "0933-7741",
publisher = "Walter de Gruyter GmbH",
number = "3",
}